9.2 Laws of Motion
9.1.For an object having mass mfalling in the zdirection,
the kinetic energy is 2 ^1 mz ̇^2 and the potential energy is mgz,
where gis the gravitational acceleration constant (approxi-
mately 9.8 m/s^2 ) and z is the position. For this one-
dimensional motion, determine the Lagrangian function Land
write the Lagrangian equation of motion.
9.2.For the system in Exercise 9.1, determine the Hamiltonian
equation of motion.
9.3.For Exercise 9.2, verify that equations 9.14 and 9.15 are
valid for the Hamiltonian you derived.
9.4. (a)A block of wood being pushed up an inclined plane
has certain forces acting on it: the force of pushing, the force
of friction, the force due to gravity. Whose equations of mo-
tion are best suited to describing this system, and why?
(b)Answer the same question but now for a rocket whose
velocity and altitude above ground are constantly being
monitored.
9.3–9.7 Unexplainable Phenomena
9.5.Draw, label, and explain the functions of the parts of a
spectroscope.
9.6.Convert (a)a wavelength of 218 Å to cm^1 , (b)a fre-
quency of 8.077
1013 s^1 to cm^1 , (c)a wavelength of
3.31 m to cm^1.
9.7.What conclusion can be drawn from the fact that two
spectra of two different compounds have certain lines at ex-
actly the same wavelengths?
9.8.Explain why no lines in the Balmer series of the hydro-
gen atom spectrum have wavenumbers larger than about
27,434 cm1. (This is called the series limit.)
9.9.What are the series limits (see the previous problem) for
the Lyman series (n 2 1) and the Brackett series (n 2 4)?
9.10.The following are the numbers n 2 for some of the se-
ries of lines in the hydrogen atom spectrum:
Lyman: 1 Balmer: 2 Paschen: 3 Brackett: 4 Pfund: 5Calculate the energy changes, in cm^1 , of the lines in each of
the stated series for each of the given values for n 1 : (a)Lyman,
n 1 5; (b)Balmer, n 1 8; (c)Paschen, n 1 4; (d)Brackett,
n 1 8; (e)Pfund, n 1 6.
9.11.Given that the wavelengths of the first three lines of the
Balmer series are 656.2, 486.1, and 434.0 nm, calculate an av-
erage value of R.
9.12.From the numbers determined by Millikan, what was
the value of the charge-to-mass ratio, e/m, in units of C/kg?
9.13. (a)Using the identities of alpha (a helium nucleus) and
beta (an electron) particles as well as the masses of the pro-
ton, neutron, and electron, estimate how many beta particles
it takes to make up the mass of one alpha particle. (b)From
this result, would you expect an alpha particle or a beta par-
ticle of the same kinetic energy to be the faster-moving ra-
dioactive emission? (c)Does your answer to part b justify the
experimental observation that beta particles are more pene-
trating than alpha particles?
9.14. (a)How much radiant energy is given off, in watt/
meter^2 , by an electric stove heating element that has a tem-
perature of 1000 K? (b)If the area of the heating element is
250 cm^2 , how much power, in watts, is being emitted?
9.15.Stefan’s law, equation 9.18, suggests that any body of
matter, no matter what the temperature, is emitting energy.
At what temperature would a piece of matter have to be in or-
der to radiate energy at the flux of 1.00 W/m^2? At the flux of
10.00 W/m^2? 100.00 W/m^2?
9.16.An average human body has a surface area of 0.65 m^2.
At a body temperature of 37°C, how many watts (or J/s) of
power does a person emit? (Understanding such emissions is
important to NASA and other space agencies when designing
space suits.)
9.17.The surface temperature of our sun is about 5800 K.
Assuming that it acts as a blackbody: (a)What is the power
flux radiated by the sun, in W/m^2? (b)If the surface area of
the sun is 6.087
1012 m^2 , what is the total power emitted
in watts? (c)Since watts are J/s, how many joules of energy
are radiated in one year (365 days)? (Note: The sun is actually
a very poor approximation of a blackbody.)
9.18.The slope of the plot of energy versus wavelength for
the Rayleigh-Jeans law is given by a rearrangement of equa-
tion 9.20:dd ^8 4 kTWhat are the value and units of this slope for a blackbody hav-
ing the following temperatures and at the following wave-
lengths? (a)1000 K, 500 nm; (b)2000 K, 500 nm; (c)2000 K,
5000 nm; (d)2000 K, 10,000 nm. Do the answers indicate
the presence of an ultraviolet catastrophe?9.19. (a)Use Wien’s law to determine the (^) maxof the sun if
its surface temperature is 5800 K. (b)The human eye sees
light most efficiently if the light has a wavelength of 5000 Å
(1 Å 10 ^10 m), which is in the green-blue portion of the
spectrum. To what blackbody temperature does that corre-
spond? (c)Compare your answers from the first two parts and
comment.
9.8 Quantum Theory
9.20.The slope of the plot of energy versus wavelength for
Planck’s law is given by a rearrangement of equation 9.22:
dd ^8 hc (^5) ehc/ k^1 T 1
Give the value and units of this slope for a blackbody having
the following temperatures and at the following wavelengths:
(a)1000 K, 500 nm; (b)2000 K, 500 nm; (c)2000 K, 5000 nm;
(d)2000 K, 10,000 nm; (e)Compare these results to those
for problem 9.18. (f)At what temperatures and spectral re-
gions will the Rayleigh-Jeans law be close to Planck’s law?
Exercises for Chapter 9 271
EXERCISES FOR CHAPTER 9